Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO...

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Chapter 15 - Oscillations

“The pendulum of the mind oscillates between sense andnonsense, not between right and wrong.”

-Carl Gustav Jung

David J. StarlingPenn State Hazleton

PHYS 211

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Oscillatory motion is motion that is periodic in

time (e.g., earthquake shakes, guitar strings).

The period T measures the time for one oscillation.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Oscillatory motion is motion that is periodic in

time (e.g., earthquake shakes, guitar strings).

The period T measures the time for one oscillation.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Oscillatory motion that is sinusoidal is known as

Simple Harmonic Motion.

x(t) = xm cos(ωt)

xm is the maximum displacementω is the angular frequency: ω = 2πf = 2π/T .

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Oscillatory motion that is sinusoidal is known as

Simple Harmonic Motion.

x(t) = xm cos(ωt)

xm is the maximum displacement

ω is the angular frequency: ω = 2πf = 2π/T .

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Oscillatory motion that is sinusoidal is known as

Simple Harmonic Motion.

x(t) = xm cos(ωt)

xm is the maximum displacementω is the angular frequency: ω = 2πf = 2π/T .

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Two oscillators can have different frequencies, or

different phases:

x(t) = xm cos(ωt + φ)

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Sinusoidal oscillations are described by these

definitions:

x(t) = xm cos(ωt + φ)

T = 1/f

ω = 2πf

I x in meters

I T in seconds

I f in Hertz (1/s)

I ω in rad/s

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Sinusoidal oscillations are described by these

definitions:

x(t) = xm cos(ωt + φ)

T = 1/f

ω = 2πf

I x in meters

I T in seconds

I f in Hertz (1/s)

I ω in rad/s

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Since we know the position of an oscillating

object, we also know its velocity and acceleration:

x(t) = xm cos(ωt + φ)

v(t) =dxdt

= −ωxm sin(ωt + φ)

a(t) =dvdt

= −ω2xm cos(ωt + φ)

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

The maximum velocity and acceleration depend

on the frequency and the maximum displacement.

Max position: xm

Max velocity: ωxm

Max acceleration: ω2xm

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Simple harmonic motion is generated by a linear

restoring force:

F = −kx = ma = md2xdt2

d2xdt2 +

km

x = 0

The solution to this differential equation:

x(t) = xm cos(√

k/mt + φ)

(so ω =√

k/m)

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Simple harmonic motion is generated by a linear

restoring force:

F = −kx = ma = md2xdt2

d2xdt2 +

km

x = 0

The solution to this differential equation:

x(t) = xm cos(√

k/mt + φ)

(so ω =√

k/m)

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

Simple harmonic motion is generated by a linear

restoring force:

F = −kx = ma = md2xdt2

d2xdt2 +

km

x = 0

The solution to this differential equation:

x(t) = xm cos(√

k/mt + φ)

(so ω =√

k/m)

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

The motion of a SHO is related to motion in a

circle.

x(t) = xm cos(ωt + φ)

v(t) = −ωxm sin(ωt + φ)

a(t) = −ω2xm cos(ωt + φ)

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

The motion of a SHO is related to motion in a

circle.

x(t) = xm cos(ωt + φ)

v(t) = −ωxm sin(ωt + φ)

a(t) = −ω2xm cos(ωt + φ)

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Simple Harmonic Oscillator (SHO)

The motion of a SHO is related to motion in a

circle.

x(t) = xm cos(ωt + φ)

v(t) = −ωxm sin(ωt + φ)

a(t) = −ω2xm cos(ωt + φ)

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Chapter 15

Lecture Question 15.1The graph below represents the oscillatory motion of threedifferent springs with identical masses attached to each.Which of these springs has the smallest spring constant?

(a) Graph 1

(b) Graph 2

(c) Graph 3

(d) Both 2 and 3 are smallest and equal

(e) All three have the same spring constant.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Energy in SHO

A spring stores potential energy. To find it,

calculate the work the spring force does:

∆U = −W = −∫ x2

x1

(−kx)dx =12

kx22 −

12

kx21

A spring compressed by x stores energy U = 12 kx2.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Energy in SHO

A spring stores potential energy. To find it,

calculate the work the spring force does:

∆U = −W = −∫ x2

x1

(−kx)dx =12

kx22 −

12

kx21

A spring compressed by x stores energy U = 12 kx2.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Energy in SHO

As a mass oscillates, the energy transfers from

kinetic to potential energy.

At the ends of the motion, velocity is zero, K is zero and Uis maximum.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Energy in SHO

As a mass oscillates, the energy transfers from

kinetic to potential energy.

At the ends of the motion, velocity is zero, K is zero and Uis maximum.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Energy in SHO

The energy oscillates between:

U =12

kx2 =12

kx2m cos2(ωt + φ)

K =12

mv2 =12

mω2︸︷︷︸k

x2m sin2(ωt + φ)

E = U + K = U =12

kx2m

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Energy in SHO

The energy oscillates between:

U =12

kx2 =12

kx2m cos2(ωt + φ)

K =12

mv2 =12

mω2︸︷︷︸k

x2m sin2(ωt + φ)

E = U + K = U =12

kx2m

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Energy in SHO

The energy oscillates between:

U =12

kx2 =12

kx2m cos2(ωt + φ)

K =12

mv2 =12

mω2︸︷︷︸k

x2m sin2(ωt + φ)

E = U + K = U =12

kx2m

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

A simple pendulum is a ball on a string. It acts

like a SHO for small angles.

The restoring force:

F = −mg sin θ ≈ −mgθ

I = mL2

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

For small angles (θ < 20◦), a pendulum is like a

simple harmonic oscillator.

τ = −(mgθ)L = Iα = Id2θ

dt2

d2θ

dt2 +mgL

I︸︷︷︸ω2

θ = 0

θ(t) = θm cos(ωt + φ)

ω =√

mgL/I =√

g/L

T = 1/f = 2π/ω = 2π√

L/g

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

For small angles (θ < 20◦), a pendulum is like a

simple harmonic oscillator.

τ = −(mgθ)L = Iα = Id2θ

dt2

d2θ

dt2 +mgL

I︸︷︷︸ω2

θ = 0

θ(t) = θm cos(ωt + φ)

ω =√

mgL/I =√

g/L

T = 1/f = 2π/ω = 2π√

L/g

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

For small angles (θ < 20◦), a pendulum is like a

simple harmonic oscillator.

τ = −(mgθ)L = Iα = Id2θ

dt2

d2θ

dt2 +mgL

I︸︷︷︸ω2

θ = 0

θ(t) = θm cos(ωt + φ)

ω =√

mgL/I =√

g/L

T = 1/f = 2π/ω = 2π√

L/g

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

For small angles (θ < 20◦), a pendulum is like a

simple harmonic oscillator.

τ = −(mgθ)L = Iα = Id2θ

dt2

d2θ

dt2 +mgL

I︸︷︷︸ω2

θ = 0

θ(t) = θm cos(ωt + φ)

ω =√

mgL/I =√

g/L

T = 1/f = 2π/ω = 2π√

L/g

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

For small angles (θ < 20◦), a pendulum is like a

simple harmonic oscillator.

τ = −(mgθ)L = Iα = Id2θ

dt2

d2θ

dt2 +mgL

I︸︷︷︸ω2

θ = 0

θ(t) = θm cos(ωt + φ)

ω =√

mgL/I =√

g/L

T = 1/f = 2π/ω = 2π√

L/g

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

For a physical pendulum with moment of inertia I

and small oscillations,

d2θ

dt2 +mgh

Iθ = 0

ω =√

mgh/I

T = 1/f = 2π/ω = 2π√

I/mgh

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

For a physical pendulum with moment of inertia I

and small oscillations,

d2θ

dt2 +mgh

Iθ = 0

ω =√

mgh/I

T = 1/f = 2π/ω = 2π√

I/mgh

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

For a physical pendulum with moment of inertia I

and small oscillations,

d2θ

dt2 +mgh

Iθ = 0

ω =√

mgh/I

T = 1/f = 2π/ω = 2π√

I/mgh

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

For a physical pendulum with moment of inertia I

and small oscillations,

d2θ

dt2 +mgh

Iθ = 0

ω =√

mgh/I

T = 1/f = 2π/ω = 2π√

I/mgh

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

A torsion pendulum is a symmetric object where

the restoring torque arises from a twisted wire.

τ = −κθ (similar to spring)

d2θ

dt2 +κ

Iθ = 0

ω =√κ/I

T = 1/f = 2π/ω = 2π√

I/κ

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

A torsion pendulum is a symmetric object where

the restoring torque arises from a twisted wire.

τ = −κθ (similar to spring)d2θ

dt2 +κ

Iθ = 0

ω =√κ/I

T = 1/f = 2π/ω = 2π√

I/κ

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

A torsion pendulum is a symmetric object where

the restoring torque arises from a twisted wire.

τ = −κθ (similar to spring)d2θ

dt2 +κ

Iθ = 0

ω =√κ/I

T = 1/f = 2π/ω = 2π√

I/κ

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Pendulums

A torsion pendulum is a symmetric object where

the restoring torque arises from a twisted wire.

τ = −κθ (similar to spring)d2θ

dt2 +κ

Iθ = 0

ω =√κ/I

T = 1/f = 2π/ω = 2π√

I/κ

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Chapter 15

Lecture Question 15.3A grandfather clock, which uses a pendulum to keepaccurate time, is adjusted at sea level. The clock is thentaken to an altitude of several kilometers. How will theclock behave in its new location?

(a) The clock will run slow.

(b) The clock will run fast.

(c) The clock will run the same as it did at sea level.

(d) The clock cannot run at such high altitudes.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

Damped simple harmonic motion is the result of

oscillatory behavior in the presence of a retarding

force.

Fd = −bv

with b the damping constant.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

Applying Newton’s Second Law to this situation,

Fnet = ma

−kx− bv = ma

0 =d2xdt2 +

bm

dxdt

+km

x

The solution:

x(t) = xme−bt/2m cos(ω′t + φ)

ω′ =

√km− b2

4m2

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

Applying Newton’s Second Law to this situation,

Fnet = ma

−kx− bv = ma

0 =d2xdt2 +

bm

dxdt

+km

x

The solution:

x(t) = xme−bt/2m cos(ω′t + φ)

ω′ =

√km− b2

4m2

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

For damped harmonic motion, the oscillations

will die out over time.

Side note: ω′ =√

km −

b2

4m2 is only valid if k/m > b2/4m2.

What happens if k/m ≤ b2/4m2?

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

For damped harmonic motion, the oscillations

will die out over time.

Side note: ω′ =√

km −

b2

4m2 is only valid if k/m > b2/4m2.

What happens if k/m ≤ b2/4m2?

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

If an oscillator of angular frequency ω is drivenby an external force at a frequency ωd, then theresponse will also be at ωd.

d2xdt2 +

km

x = F0 cos(ωt)→ x(t) = A cos(ωt + φ)

The amplitude A depends on the relationship between ωd

and ω.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

If an oscillator of angular frequency ω is drivenby an external force at a frequency ωd, then theresponse will also be at ωd.

d2xdt2 +

km

x = F0 cos(ωt)→ x(t) = A cos(ωt + φ)

The amplitude A depends on the relationship between ωd

and ω.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

If an oscillator of angular frequency ω is drivenby an external force at a frequency ωd, then theresponse will also be at ωd.

d2xdt2 +

km

x = F0 cos(ωt)→ x(t) = A cos(ωt + φ)

The amplitude A depends on the relationship between ωd

and ω.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

If a damped oscillator is driven at its naturalfrequency, the system is on resonance and theoscillations are maximum.

The width of the resonance peak depends on the dampingconstant.

Chapter 15 - Oscillations

Simple HarmonicOscillator (SHO)

Energy in SHO

Pendulums

Damped Oscillations

Damped Oscillations

If a damped oscillator is driven at its naturalfrequency, the system is on resonance and theoscillations are maximum.

The width of the resonance peak depends on the dampingconstant.